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Holographic Plasmon Relaxation with and Without Broken Translations IFT-UAM/CSIC-19-54 Holographic plasmon relaxation with and without broken translations Matteo Baggiolia , Ulf Granb , Amadeo Jimenez Albaa , Marcus Torns¨ob , Tobias Zinggc aInstituto de Fisica Teorica UAM/CSIC, c/ Nicolas Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain bDepartment of Physics, Division for Theoretical Physics, Chalmers University of Technology SE-412 96 G¨oteborg, Sweden cNordita, Stockholm University and KTH Royal Institute of Technology Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We study the dynamics and the relaxation of bulk plasmons in strongly coupled and quantum critical systems using the holographic framework. We analyze the dispersion relation of the plasmonic modes in detail for an illustrative class of holographic bottom-up models. Comparing to a simple hydrodynamic formula, we entangle the complicated interplay between the three least damped modes and shed light on the underlying physical processes. Such as the dependence of the plasma frequency and the effective relaxation time in terms of the electromagnetic coupling, the charge and the temperature of the system. Introducing momentum dissipation, we then identify its additional contribution to the damping. Finally, we consider the spontaneous symmetry breaking (SSB) of translational invariance. Upon arXiv:1905.00804v2 [hep-th] 18 Aug 2019 dialing the strength of the SSB, we observe an increase of the longitudinal sound speed controlled by the elastic moduli and a decrease in the plasma frequency of the gapped plasmon. We comment on the condensed matter interpretation of this mechanism. Contents 1 Introduction1 2 The model 7 3 The effects of the electromagnetic interactions9 4 Holographic plasmons at finite density 12 5 Plasmons with broken translations 18 6 The effects of the spontaneous breaking of translational invariance 21 7 Conclusions 27 A Equations of motion 29 A.1 Linear axions 29 A.2 SSB model 31 B Numerical techniques 32 1 Introduction Plasma oscillations, whose quanta are called plasmons, are collective electronic oscillations in metals. Their history is an incredible and successful connubium between art and technology. Already in the 4th century, Romans manufactured dichroic glass, used for example in the Ly- curgus Cup, where plasmon effects from gold and silver particles dispersed in the glass matrix interplay with transmitted and reflected light at certain wavelengths. These techniques in staining glass where later refined and became one of the major artistic techniques in the Mid- dle Ages [1], where stunning artworks with stained glass were created, c.f. figure1. Although plasmonic effects have been known for over a millennium, a complete understanding of the underlying physics of these phenomena was not accomplished until the 1970s, which marks the beginning of modern plasmon-based applications. The technological uses of plasmons { 1 { have been motivated by the attempt to overcome the diffraction limit of light and by their ability to highly enhance the electric field intensity. The modern applications are limitless, from solar cell and cosmetics to a completely new branch of science known as plasmonics [2,3]. Plasmons appear as a result of interaction processes between electromagnetic radiation and conduction electrons and their main features can be understood with classical electro- magnetism [4]. The dispersion relation of the longitudinal plasmon modes is characterized by the longitudinal part of the (tensorial) dielectric function = @Di { see e.g. [5] for details. ij @Ej More precisely, the condition that a plasmon excitation leads to an oscillation in polarization P without the influence of a change in external field D leads to the condition L(k; !) = 0 ; (1.1) which we therefore will call `plasmon condition' in the following. Using Maxwell's equations in a medium and basic definitions, the dielectric function can also be shown to be related to other elementary properties, like the conductivity of the material i 4 π σL(k; !) L(k; !) = 1 + ; (1.2) ! 0 where 0 is the vacuum dielectric constant and σL(k; !) the spatially resolved longitudinal conductivity. Over a wide range of frequencies, the optical properties of metals can be ex- plained using the free electrons gas approximation. This model idealizes the physics of a metal considering it as a gas of free electrons with number density n moving against a fixed background of positive ion cores. The electrons are driven by an applied external electromag- netic field, and their motion is damped with a characteristic collision frequency γ = 1/τ . The timescale τ is labelled as the relaxation time of the free electron gas. For a free electron gas it is then a standard calculation to derive the dielectric function1 [5], !2 (!) = 1 − p ; (1.3) !2 + i γ ! where, crucially, we define the quantity 2 2 4 π n e !p = ∗ ; (1.4) m 0 as the plasma frequency of the metal. Here we use the symbol m∗ to denote the effective mass of the electronic quasi-particles, which can differ significantly from the microscopic electron mass2. For large frequencies !τ 1, we can neglect the damping effects and the dielectric function is purely real: !2 (!) = 1 − p ; (1.5) !2 1Here, for simplicity, we omit the sub-index L indicating that the dielectric function we are after relates to the longitudinal collective modes of the system. 2Let us already mention that no quasi-particles are present in the holographic picture we discuss in this paper. Therefore, this weakly coupled logic is a good guidance but it should not be taken too seriously. { 2 { ω/ωp 2 plasmon 1 light k c/ωp 1 2 Figure 1: Left: One example of the applications of plasmons in art in the form of a stained glass window in Notre Dame, Paris. Right: The typical bulk plasmon dispersion relation 2 2 2 2 ! = !p + c k compared with the photon dispersion relation in vacuum. i.e. the known result for the underdamped free electron plasma (without taking into account interband transitions). In the other limit, !τ 1, the imaginary part of the dielectric function cannot be discarded and the physics is very different. In this last regime, the metals are mainly absorbing and the fields fall off inside the sample following Beer's law ∼ e−z/δ, where δ is known as the skin-depth. As we will see later, these features can be linked to the so-called k-gap dispersion relation [6] which will appear in our model as well. The nature of plasmons in ordinary, and weakly coupled, metals (Fermi liquids) can be understood using rather simple classical models [5] and it can be summarized simply as the presence of a sound mode dressed by electromagnetic interactions. It is important to clarify that the sound mode we are discussing is the electronic sound, usually denoted as zero sound. Strictly speaking, this term refers to the sound mode related to the zero temperature shape deformations of the Fermi surface. This hydrodynamic mode can be obtained using Fermi liquid theory [7] and it is just a manifestation of the elastic property of the electronic liq- uid [8]. Here, we use the term \zero sound\ in a broader sense, to simply define the collective sound mode of the quantum critical soup both at zero and finite temperature. Despite the similar nature, it is not the normal sound associated with the vibrational modes of the ionic lattice, i.e. the standard phonons. We will discuss this point further in section6. The underlying picture is less clear when more \exotic" phases are considered. In particular, recent experiments [9{13] examined the dynamics of the collective plasmon modes in strongly correlated and quantum critical materials. The results are quite different from the weakly cou- pled paradigm, in the sense that the plasmon modes display an anomalously strong damping. { 3 { Not only that, but there are recent observations [14, 15] which indicate that when increas- ing the momentum, plasmons stop existing as well-defined quasi-particles and get smoothed out in the collective and incoherent \quantum soup" losing a characteristic momentum scale. The strong damping of plasmons, even at zero momentum, is expected for strongly coupled quantum critical systems and was first observed in a holographic model in [16]. This was later elaborated on in [17], which promoted the idea that the plasmons are no longer kinematically protected, since the electron-hole (Lindhard) continuum is now substituted by a continuum of modes typical of quantum critical systems. As a consequence, the plasmons can easily decay into such incoherent set of states and therefore display an anomalous and new damping mechanism (even at zero momentum). The previous discussion leads us to an important point of this paper which is the identi- fication and the analysis of the dielectric relaxation and plasmons damping mechanisms in a strongly coupled medium, as e.g. in a quantum critical phase of matter. Working with linear response, and assuming zero momentum for simplicity, the complex dielectric function (!) controls the relation between the electric field E and the polarization P , P (!) = 0 [(!) − 1] E(!) ; (1.6) Writing this relation in position space, Z P (t) = 0 [(t − s) − δ(t − s)] E(s) ds ; (1.7) it becomes manifest that a non-trivial dielectric function implies a delay, i.e. relaxation mech- anism, between the electric field and the polarization in the material. The simplest phenomenological model for the dielectric constant is the well-known Debye model. It relies on a single relaxation time approximation for the polarization: dP (t) 1 = − P (t) ; (1.8) dt τD where the timescale τD is denoted as the Debye relaxation time. Within this model, the dielectric function can be written simply as: ∆ (!) = 1 + ; (1.9) 1 + i ! τD where 1 is its value at infinite frequency.
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